First derivatives at the optimum analysis (\textit{fdao}): An approach to estimate the uncertainty in nonlinear regression involving stochastically independent variables

An important problem of optimization analysis surges when parameters such as $ \{\theta_j\}_{j=1,\, \dots \,,k }$, determining a function $ y=f(x\given\{\theta_j\}) $, must be estimated from a set of observables $ \{ x_i,y_i\}_{i=1,\, \dots \,,m} $. Where $ \{x_i\} $ are independent variables assumed to be uncertainty-free. It is known that analytical solutions are possible if $ y=f(x\given\theta_j) $ is a linear combination of $ \{\theta_{j=1,\, \dots \,,k} \}.$ Here it is proposed that determining the uncertainty of parameters that are not \textit{linearly independent} may be achieved from derivatives $ \tfrac{\partial f(x \given \{\theta_j\})}{\partial \theta_j} $ at an optimum, if the parameters are \textit{stochastically independent}.